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Monday, December 01, 2008

What if...

We've been wondering what to do this year at advent time. Last year's “Plottl a Day” was great fun, but this year unfortunately neither Stefan nor I have much time. Thus, we thought this year we'd make it an interactive advent season and count on you to make it memorable!

We will suggest a one-line science fiction scenario every day, and hope you let your fantasy go wild :-)

Here is the first:

What if we could change the value of Planck's constant?

Also... we haven't yet quite 24 “What if's” together. So, if you have a scenario that you'd love to speculate about, leave it in the comments!

18 comments:

This is simple: Nothing changes. It's a dimensionful constant. You can for example undo a change of hbar by a scale transformation. Or you should say what you want to keep fixed.

What you are probably thinking of is keeping the Bohr radius (setting the size of macroscopic matter) fixed while changing hbar (while most people usually think of the Bohr radius as a function of other constants including hbar). In that case we would see more quantum effects in the macroscopic world. It's hard to imagine what such a world would look like. You might be tempted to thing everything looks a bit blurry but I am not sure that is the case. Even if I don't know the exact position of my coffee mug I know there is exactly one and a world where there is none or two (two incarnations at different positions) is not realised.

It's probably save to say that all effects would be of interference type: If I allow my particles more options (like opening the second slit in a double slit experiment), the counting at certain positions goes down (rather than up with classical probabilities when more options are allowed.

Robert is roughly correct, in the sense that it is hard to define changing a specific "constant." That is due to the question: what is left that defines the standard for calling the parameter changed and by how much? In the case of light speed, other things like Bohr radius and wavelength relative to energy would change.

However, we can easily imagine some parameters changing but "everything else the same." One obvious example is G: it can be altered and leave most physics alone enough to define the change (same atomic physics but less strength of gravity.) We can certainly change h in the sense that wavelength to energy or momentum relation changes, assuming we have other ways of defining size and mass. Note we can imagine specific effects of "all mass is doubled" etc. if we keep other constants the same, and have reference entities like electrons to work with.

George Gamow tried to imagine c and h being very different in various installments of "Mr. Tompkins" stories. It is interesting that despite apparent validity of formal arguments against meaningful "change", for many "constants" the Tompkins works show just what we intuitively expect from a very slow speed of light, a very large Planck's constant, etc.http://en.wikipedia.org/wiki/Mr_Tompkins

One thing I wonder is, if we could make huge nuclei Z = 1,000 and such stable, what would they be like?

Z=1000 is a lot. There are certain arguments that suggest that those would be very different, basically, they would be small neutron stars. The mathematical physicists that I am currently interacting with (teaching mathematical quantum mechanics) discuss this under the key word "stability of matter", see google.

Roughly, the idea is as follows: Why are ordinary atoms stable (have a ground state) although the 1/r Coulomb potential is unbounded from below? Well, if you want to gain a lot from the potential energy, say E, then at least one electron would have to be at about r=1/E. But such small r implies a momentum p=hbar/r = hbar E via the uncertainty relation (Sobolev inequality for the mathematicians). Such a momentum brings with it a kinetic energy p^2/2m = hbar^2 E^2/2m . Thus when you try to lower the potential energy, the kinetic energy grows qudratically and outruns the gain in potential energy.

That was the non-relativistic story. But at small r you should actually use a relativistic version of the argument and at large p the relativistic kinetic energy is linear rather than quadratic in momentum. In that case potential and kinetic energy in the above argument grow with the same power of r and it depends on the precise coefficients to determine if the total energy grows or falls in this limit.

This is where Z comes in. For this argument, Z is the relative coefficient between the potential and kinetic term and for large Z the potential energy wins. Careful calculations suggest that Z about 137 (yes, the fine structure constant comes in here) is the critical value and above it pays in terms of total energy for the electrons to be at small r.

Of course, this is a simplified model that assumes the Coulomb field is a non-dynamical background and the source is pointlike. But the simple argument holds as long as these assumptions are not too wrong. Thus we can expect that the electrons collapse to a radius where at least it is energetically possible to create new electron-positron pairs or where the potential ceases to be of 1/r form, which happens at r at the order of the radius of the nucleus.

Thus, if there were nuclei with Z>137 the size of the electron cloud would jump from being of the order of a Bohr radius to the size of the nucleus and the energy would be of the order of the potential energy at r=r_nucleus larger, many order of magnitude more than the few eV's typical of atomic physics.

Interesting, I would say if Pi were a rational number reality in terms of space would have to be discrete, for as Cantor proved it is only the irrationals that can form an uncountable infinity and therefore capable of representing an unbroken and inseparable whole. If this were true then as space expanded it would become increasingly dilute and revealing itself ever more so as being discrete. The question then would be to ask what one imagined to be what existed in between?

You probably have 24 ready by now, but I have a "what if" for people to speculate about: what if we found a fundamental theory of physics that was directly derived from number theory, or a similarly fundamental mathematics.

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